Excercise about a solar-like system

An Earth-like planet orbits around a Sun-like star with a circular orbit of period T=1 year. The system is very far from us. Assuming that our Earth is on orbit plane of the Earth-like planet, calculate:

i) the period of the partial occultation of the star from the planet;

ii) the resulting apparent magnitude variation of the star.

iii)Repeat the exercise for a giant planet, Jupiter-like, with an orbital period of T=4332 days, placed in the same system;

iv) Are the above mentioned phenomena observable with a ground-telescope?

i) 1 year
ii)Explain what is Rsun and Rearth .
you need apparent mag variation not luminos variation i think the ratio of fluxes σΤ^4 is equal to the ratio of appar magnitudes.
luminosity of star is always the same,but we can suppose it emits ration from a smaller surface
appar mag before/appar mag after = (L1/4πd ) / (L2/4πd)

You have to find the percentage of covered disk and say L2= L1/y

would be good approximation to subtract πRsun - πRearth = S' sun

and say S' sun /S sun = L 2 /L1 hence the ratio L2/L1 is known which means ratio of appar magnitudes is also known.

The question until now was extremely stupid which shows that the person who created is unlikely knows from astronomy ,amnd the last part requires knowledge that only kepler space telescope can detect earty like planets.Resolutions of ground based telescopes i think (with or withourt adaptive optics i cant remewmber) are 1'' .

when you subtract the apparent diam of the planet maybe you should take into account the distance of the planet from the star but then again if we suppose that the star is ligh years away from us ?i fthe exercise has given the distances of the planets from the stars you should find way to calculate the new apparent surfaces of the planets and subtr the from the Sun surface area.maybe you can use inverse square law

You say: You have to find the percentage of covered disk and say L2= L1/y
and I did that, and in fact this percentage depends on the covering planet radius.
You say: would be good approximation to subtract πRsun - πRearth = S' sun
but I really don't understand the general meaning of you explanation...

Let's start from the start with your expression α= 2 arcsin (REarth + RSun / 1 U.A. ). Obviously there are some parentheses missing here. What I don't see is how you got the value 12.9 hours. Is that the value you know is correct? Using your numbers (Earth radius = 6370 km; Sun radius = 694000 km; 1 U.A. = 1.49 x 108 km) results in 13.1 hours. A value of 12.9 hours corresponds to using (Rsun-Rearth)/1 AU.

Which value is correct depends on what you are supposed to be measuring.

There is a slightly easier way to do this. You don't need arcsin. The small angle approximation is valid to more than four decimal places here.

The same concept of approximation applies to the variation in magnitude. You could use the Stefan-Boltzmann law in full to get the answer. The Earth-like (or Jupiter-like) planet is also radiating, so the "correct" answer should account for this radiation. However, since radiation is proportional to T4, you can ignore the radiation from the planet itself (the relative error that results from this simplification is on the order of 10-6 or so). That leaves you with a straightforward calculation that does not involve temperature at all for the reduction in intensity.

There's one problem here: The problem did not ask for the reduction in intensity. It asked for the "resulting apparent magnitude variation of the star". You need to convert that reduction in intensity to a reduction in apparent magnitude.

Thanks.
I repeated the calculation about the eclipse period, and it results in 12.9 hours again.
The α angle is equal to 9.3 x 10-3 radiants for you too?

However, I never thought about the possibility to "adding" the planet emission to the star emission.
In my opinion, it's natural to think about a decrease in luminosity caused from the planet transit.
(When we have a solar eclipse here on the earth we see the dark and not a "powerized sun").

You say that the calculus for the intensity is straightforward, but it's just what i can not understand, as you can see from my formula about the luminosity...

Regarding the ellipse period: Show your calculations. It appears you might be losing precision due to some intermediate result.

Regarding luminosity: You missed the point. To be pedantically correct one should incorporate the planetary emissions. However, this pedantic correctness will only affect the result in the sixth decimal place. Why bother? A real telescope, earth-based or space-based, will not see that error. One important aspect of physics is learning to distinguish between that which is important and that which isn't.

Finally, regarding apparent magnitude versus luminosity, your text or lectures must have something to say about magnitude and how it relates to luminosity. How do you calculate apparent magnitude?

Of course It can not.
But I don't know the limit, or the law, that governs the magnitude limit parameter.
I only know that 10-4 is a really low difference in magnitude.
Maybe we can say that it equals to a difference in luminosity equal to 1%, but this can not help me neither...